The first number tells us that there is 1 way to assign the finishing places of the race if there is 0 ties (1 2 3 4), which has 4! = 24 possible outcomes.

The second number tells us that there are 2 ways to assign the finishing places of the race if 2 horses tie (2 2) and (2 1 1) which has 2!/2!(4!/2!2!) + 3!/2!1!(4!/2!1!1!) = 1(6) + 3(12) = 42 possible outcomes.

The third number tells us that there is 1 way to assign the finishing places to the race if 3 horses tie (3 1) which has 2!/1!1!(4!/3!1!) = 2(4) = 8 possible outcomes.

The fourth number tells us that there is 1 way to assign the finishing places to the race if 4 horses tie (4) which has 1!/1!0! = 1 outcome.

Which gives us a total of 24 + 42 + 8 + 1 = 75 outcomes.

I was wondering if one of you two can notice a pattern in the triangle that will give us the next line.

I did notice bobbym that you did some research and noted that a closed form is not likely.

Would this indicate that there is not a pattern that we could find to predict the next line in this triangle?